In his article in the January 1991 issue of SIAM News, "EuclideanGeometry Alive and Well," Barry Cipra describes three recent results inEuclidean geometry.

The first, an optimization problem called the Steiner ratio conjecture,was solved by Ding Zhu Du and Frank Hwang. The Steiner ratio conjectureinvolves objects called trees, which are line segments that connect agiven (finite) set of points, without ever looping back on itself. Thus,in a tree, if you follow a path, you can never get back to your startingpoint, or even to a point you have already moved through. A tree can bethought of as an optimization problem: how can you connect all the dotswith the shortest length of string? Obviously, one thing you don't wantto do is double back on yourself (this would be redundant), and that isexactly what having a tree insures will not happen. The question thenbecomes: What is the shortest tree one can have connecting all the dots?

But this is not exactly the question that the Steiner ratio conjectureanswers. The Steiner ratio conjecture has to do with the ratio of theshortest tree connecting a set S of points, and the shortest Steiner treeconnecting the same set S of points. Steiner trees are different from"normal" trees because in connecting the dots, you can add extra dots.Obviously, normal trees are just special cases (no extra points) ofSteiner trees. Why add the extra points? They can make the tree lengthshorter, as this diagram illustrates.

a a | /\

| / \ | d / \ / \ / \ / \ / \ / \ / \ b c b c

Let's say that abc is an equilateral triangle with sides of length 1. TheSteiner tree on the right, formed by adding the center of the triangle,d, as a point, is shorter than the tree using only points a, b, and c.The normal tree has length 2; the Steiner tree has length Sqrt(3). Itturns out that this ratio, Sqrt(3)/2, is the smallest that the ratio canbe, provided that you are measuring the shortest Steiner tree and theshortest regular tree possible (these do exist).

The second result that Cipra discusses is an algorithm for triangulatinglarge polygons (large meaning many (n) vertices). Bernard Chazelle hasfound an algorithm that will triangulate (cut up into triangles, withcertain rules about what kind of intersections the triangles can have) apolygon in an amount of time proportional to the number of vertices thatthe polygon has. Since a more naive approach yields a time proportionalto the square of the number of vertices, this algorithm makes it muchless time consuming for a computer to triangulate a polygon.

The third result Cipra summarizes is the long-held conjecture that thebest way to stack spheres is the "face centered cubic lattice packing"or, in layman's terms, the way grocers stack oranges. Wu-Yi Hsiang provesthis problem that Kepler proposed in 1611. However,as the problem hassuch a long history, mathematicians will be going over Hsiang's proofwith a fine tooth comb, and, right now at least, the jury is not in onwhether or not Hsiang's proof is valid.

For further details see "Euclidean Geometry Alive and Well," by Barry A.Cipra, in the January 1991 issue of SIAM News.